{"paper":{"title":"Simplicial complexes and Macaulay's inverse systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"Adam Van Tuyl, Fabrizio Zanello","submitted_at":"2007-12-11T19:47:15Z","abstract_excerpt":"Let $\\Delta$ be a simplicial complex on $V = \\{x_1,...,x_n\\}$, with Stanley-Reisner ideal $I_{\\Delta}\\subseteq R = k[x_1,...,x_n]$. The goal of this paper is to investigate the class of artinian algebras $A=A(\\Delta,a_1,...,a_n)= R/(I_{\\Delta},x_1^{a_1},...,x_n^{a_n})$, where each $a_i \\geq 2$. By utilizing the technique of Macaulay's inverse systems, we can explicitly describe the socle of $A$ in terms of $\\Delta$. As a consequence, we determine the simplicial complexes, that we will call {\\em levelable}, for which there exists a tuple $(a_1,...,a_n)$ such that $A(\\Delta,a_1,...,a_n)$ is a le"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0712.1804","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}